With a dressage saddle you are safer when you are experienced enough to be ‘in the saddle’ or ‘down in the saddle,’ or ‘not top heavy’.
This means if the horse shies you go with it and you are not left behind where the horse was when it shied. This comes with riding experience and why you will hear western riders call an English saddle an ejection seat. In our case the western saddles did not have a horn like American Western saddles, they used Australian Stock saddles with ears above your thighs. They also ride with their feet at their shoulders in a chair position.
With the trainer above, riding a Prix St George horse, grabbing a pommel or grab strap, we have no idea why. They would long ago have been experienced enough not to do that on lower level horses. Now they are older and riding an upper level lofty horse. It could be because of an injury, because of age, because of pain, because of fear, because the rider does not have the core muscle for the level they are riding, because they are riding a horse with a terrible trot that none if us could ride either without grabbing the pommel, because their body won’t do what it used to, or a myriad of other problems we dont know of.
As to the rule - in the Dressage Judge lectures I have been to, this has never been discussed and I have never seen it in group lessons above beginners or tests I have judged, so it is not a common thing to do in dressage in my neck of the woods, which is Australia.
This could be that most people in Australia own their own horse/s and keep them at home and ride more out of lessons that they do in lessons, so they have a lot more hours in the saddle than someone who only rides in lessons.